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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><br/>  for <font color="Olive" title="b2">Ka</font> being   non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Olive" title="b1">V</font>  st <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b2">Ka</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b2">Ka</font> holds <br/> <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b2">Ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">Ka</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Ka</font> being   non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c2">V</font>  st <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Olive" title="b1">Ka</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">Ka</font> holds <br/> <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">Ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">Ka</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Ka</font> be   non  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>   <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>  <a href="simplex1.html#NM2" title="SIMPLEX1:NM.2">SimplicialComplex</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">Ka</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Ka</font> implies  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">Ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Ka</font>)</span>)</span> )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>(  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">Ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> ($1,<font color="Maroon" title="c3">Ka</font>)</span>)</span> &amp;  not  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> ($1,<font color="Maroon" title="c3">Ka</font>) is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp;  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> ($1,<font color="Maroon" title="c3">Ka</font>) is  <a href="simplex1.html#V1" title="SIMPLEX1:attr.1">affinely-independent</a>  );<br/>
<span class="kw">assume </span><a NAME="E1:58"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">Ka</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Ka</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">Ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Ka</font>)</span>)</span></span><br/>

<a NAME="E2:58"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<a NAME="E2:58_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">Ka</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">Ka</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="simplex1.html#T18" target="_self" title="SIMPLEX1:th.18">Th18</a></span>;<br/>
<a NAME="E3:58_1"/>
(  <a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>,<font color="Maroon" title="c3">Ka</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex1.html#K4" title="SIMPLEX1:func.4">BCS</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">Ka</font>)</span>)</span> &amp; <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">Ka</font>)</span>)</span></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="simplex1.html#K3" title="SIMPLEX1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">Ka</font></span><a href="simplex1.html#K3" title="SIMPLEX1:func.3">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="simplex1.html#T10" target="_self" title="SIMPLEX1:th.10">Th10</a>, <a class="ref" href="simplex1.html#T20" target="_self" title="SIMPLEX1:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:58_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E1:58_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E2:58_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="simplex1.html#T28" target="_self" title="SIMPLEX1:th.28">Th28</a>, <a class="ref" href="simplex1.html#T31" target="_self" title="SIMPLEX1:th.31">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:58"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="simplex1.html#T16" target="_self" title="SIMPLEX1:th.16">Th16</a></span>;<br/>
<a NAME="E4:58"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E37">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:58"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">Ka</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <span class="p1">(<span class="default"><a href="simplex1.html#K5" title="SIMPLEX1:func.5">BCS</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">Ka</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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